Noncommutative Trigonometry

Trigonometry of Quantum States

“…See the account in the book [3], and for more recent developments, the survey paper [5]. The min-max result states, and let us here restrict to finite dimensional selfadjoint (e.g., symmetric or Hermitian) positive operators A such as those we will treat in this paper, that…”

“…The operator trigonometry has other facets of course and a number of interesting applications to numerical analysis, statistics, Bell's inequalities, among others. We refer the reader to [3][4][5] and other papers we may cite in the course of this paper, for further information. But to help the reader for whom my operator trigonometry is new, let me mention three useful insights.…”

“…Using this 6 × 6 example, we point out that one could also consider the higher antieigenvalues and the corresponding smaller internal turning angles (see [5,12]) for ρ AB and its partial transpose with respect to A and the other examples treated in [11], should those be of interest. For example the subspace turning angle of the partial transpose matrix corresponding to the eigenvalues 1/4 and 1/8 is φ(1/4, 1/8) = sin −1 (1/3), no matter what the value of the noise coupler x.…”

“…And the average turning angles are not really the actual geometrical critical turning angles of the operator. There is much more information in all the internal turning angles defined by my higher antieigenvalues and higher antieigenvectors [5,12].…”

The Trigonometry of Matrix Statistics

“…Further theoretical background for the matrix trigonometry may be found in the books Gustafson (1997), Gustafson & Rao (1997), and the papers cited in the bibliography here. Also a recent review is given in Gustafson (2006). The reader is referred to those sources for the terminology (all due to this author) to be used in this exposition.…”